Independent Sets in Graphs with an Excluded Clique Minor

Let $G$ be a graph with $n$ vertices, with independence number $\alpha$, and with with no $K_{t+1}$-minor for some $t\geq5$. It is proved that $(2\alpha-1)(2t-5)\geq2n-5$

On Multiplicative Sidon Sets

Fix integers $b>a\geq1$ with $g:=\gcd(a,b)$. A set $S\subseteq\mathbb{N}$ is \emph{$\{a,b\}$-multiplicative} if $ax\neq by$ for all $x,y\in S$. For all $n$, we determine an $\{a,b\}$-multiplicative set with maximum cardinality in $[n]$, and conclude that the maximum density of an $\{a,b\}$-multiplicative set is $\frac{b}{b+g}$. For $A, B \subseteq \mathbb{N}$, a set $S\subseteq\mathbb{N}$ is \emph{$\{A,B\}$-multiplicative} if $ax=by$ implies $a = b$ and $x = y$ for all $a\in A$ and $b\in B$, and $x,y\in S$. For $1 < a < b < c$ and $a, b, c$ coprime, we give an O(1) time algorithm to approximate the maximum density of an $\{\{a\},\{b,c\}\}$-multiplicative set to arbitrary given precision

The "Double Sense" of Fichte's Philosophical Language - Some Critical Reflections on the Cambridge Companion to Fichte

The principal thesis in this review-essay is that the key linguistic terms in Fichte’s Wissenschaftslehre especially have two main meanings that appear at first sight to be almost in contradiction or opposed to each other. The reader of Fichte therefore has to work hard to overcome any apparent conflicts in the “double sense” of his philosophical terminology. Accordingly, I argue that Fichte’s linguistic method and use of language should be seen as part of his chief philosophical method of synthesis, where we have to carry out a similar procedure and attempt to reconcile opposites using the power of the imagination. This thesis is put forward by means of a number of practical examples and in the context of some critical reflections on the recently published Cambridge Companion to Fichte, eds. David James and Günter Zöller. Review essay published in Volume 15 (December, 2017) of the journal Revista de Estud(i)os sobre Fichte (ed. Emiliano Acosta)

Colouring the Square of the Cartesian Product of Trees

We prove upper and lower bounds on the chromatic number of the square of the cartesian product of trees. The bounds are equal if each tree has even maximum degree

On Tree-Partition-Width

A \emph{tree-partition} of a graph $G$ is a proper partition of its vertex set into `bags', such that identifying the vertices in each bag produces a forest. The \emph{tree-partition-width} of $G$ is the minimum number of vertices in a bag in a tree-partition of $G$. An anonymous referee of the paper by Ding and Oporowski [\emph{J. Graph Theory}, 1995] proved that every graph with tree-width $k\geq3$ and maximum degree $\Delta\geq1$ has tree-partition-width at most $24k\Delta$. We prove that this bound is within a constant factor of optimal. In particular, for all $k\geq3$ and for all sufficiently large $\Delta$, we construct a graph with tree-width $k$, maximum degree $\Delta$, and tree-partition-width at least (\eighth-\epsilon)k\Delta. Moreover, we slightly improve the upper bound to ${5/2}(k+1)({7/2}\Delta-1)$ without the restriction that $k\geq3$

Drawing a Graph in a Hypercube

A $d$-dimensional hypercube drawing of a graph represents the vertices by distinct points in $\{0,1\}^d$, such that the line-segments representing the edges do not cross. We study lower and upper bounds on the minimum number of dimensions in hypercube drawing of a given graph. This parameter turns out to be related to Sidon sets and antimagic injections.Comment: Submitte

Partitions and Coverings of Trees by Bounded-Degree Subtrees

This paper addresses the following questions for a given tree $T$ and integer $d\geq2$: (1) What is the minimum number of degree-$d$ subtrees that partition $E(T)$? (2) What is the minimum number of degree-$d$ subtrees that cover $E(T)$? We answer the first question by providing an explicit formula for the minimum number of subtrees, and we describe a linear time algorithm that finds the corresponding partition. For the second question, we present a polynomial time algorithm that computes a minimum covering. We then establish a tight bound on the number of subtrees in coverings of trees with given maximum degree and pathwidth. Our results show that pathwidth is the right parameter to consider when studying coverings of trees by degree-3 subtrees. We briefly consider coverings of general graphs by connected subgraphs of bounded degree